4.1.1.0.2 Example 2

\begin{align} x^{\prime }\left ( t\right ) +y^{\prime }\left ( t\right ) & =x+y+t\tag {1}\\ 2x^{\prime }\left ( t\right ) +y^{\prime }\left ( t\right ) & =2x+3y+e^{t} \tag {2}\end{align}

Let \(x^{\prime }=A,y^{\prime }=B\) then

\begin{align} A+B & =x+y+t\tag {1}\\ 2A+B & =2x+3y+e^{t} \tag {2}\end{align}

From (1), \(B=x+y+t-A\). Substituting in (2) gives

\begin{align} 2A+\left ( x+y+t-A\right ) & =2x+3y+e^{t}\nonumber \\ A & =x-t+2y+e^{t} \tag {3}\end{align}

Now we plugin the above in (1) which gives

\begin{align} \left ( x-t+2y+e^{t}\right ) +B & =x+y+t\nonumber \\ B & =2t-y-e^{t} \tag {4}\end{align}

Hence we have the following two linear ode’s of standard form now. These are (3,4)

\begin{align*} x^{\prime } & =x-t+2y+e^{t}\\ y^{\prime } & =2t-y-e^{t}\end{align*}

And now these can be solved using standard methods.